Line Element Geometry for 3D Shape Understanding and ...

Line Element Geometry for 3D ShapeUnderstanding and ReconstructionM. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. SteinerGeometric Modeling and Industrial GeometryVienna University of TechnologyNovember 23, 2005M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Task we want to solve . . .◮ Given: data set of a 3D shape◮ Goal: understand and reconstruct the special geometry◮ Tools: line elements and their relation to equiform kinematicsM. Hofer, B. Odehnal, H. Pottmann, T. Steiner, J. Wallner: 3Dshape understanding and reconstruction based on line elementgeometry. Proc. ICCV’05, Vol. 2:1532–1538, 2005.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Task we want to solve . . .◮ Given: data set of a 3D shape◮ Goal: understand and reconstruct the special geometry◮ Tools: line elements and their relation to equiform kinematicsM. Hofer, B. Odehnal, H. Pottmann, T. Steiner, J. Wallner: 3Dshape understanding and reconstruction based on line elementgeometry. Proc. ICCV’05, Vol. 2:1532–1538, 2005.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Task we want to solve . . .◮ Given: data set of a 3D shape◮ Goal: understand and reconstruct the special geometry◮ Tools: line elements and their relation to equiform kinematicsM. Hofer, B. Odehnal, H. Pottmann, T. Steiner, J. Wallner: 3Dshape understanding and reconstruction based on line elementgeometry. Proc. ICCV’05, Vol. 2:1532–1538, 2005.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Task we want to solve . . .◮ Given: data set of a 3D shape◮ Goal: understand and reconstruct the special geometry◮ Tools: line elements and their relation to equiform kinematicsM. Hofer, B. Odehnal, H. Pottmann, T. Steiner, J. Wallner: 3Dshape understanding and reconstruction based on line elementgeometry. Proc. ICCV’05, Vol. 2:1532–1538, 2005.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Equiform kinematics & velocity vector fields◮ An equiform transformationx ↦→ y = αAx + a, α > 0, A ∈ SO 3 , a ∈ R 3 .is a rigid body motion together with a scaling.◮ For a smooth equiform motion, α, a, A depend smoothly on atime parameter t. Velocity vectors of points are given byẏ(t) = ( ˙αA + αȦ)x + ȧ = · · · = c × y + c + γy,with vectors c, c and a real number γ.◮ Use 7-tuple (c, c, γ) to encode velocity vector fields whichoccur with smooth equiform motions.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Equiform kinematics & velocity vector fields◮ An equiform transformationx ↦→ y = αAx + a, α > 0, A ∈ SO 3 , a ∈ R 3 .is a rigid body motion together with a scaling.◮ For a smooth equiform motion, α, a, A depend smoothly on atime parameter t. Velocity vectors of points are given byẏ(t) = ( ˙αA + αȦ)x + ȧ = · · · = c × y + c + γy,with vectors c, c and a real number γ.◮ Use 7-tuple (c, c, γ) to encode velocity vector fields whichoccur with smooth equiform motions.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Equiform kinematics & velocity vector fields◮ An equiform transformationx ↦→ y = αAx + a, α > 0, A ∈ SO 3 , a ∈ R 3 .is a rigid body motion together with a scaling.◮ For a smooth equiform motion, α, a, A depend smoothly on atime parameter t. Velocity vectors of points are given byẏ(t) = ( ˙αA + αȦ)x + ȧ = · · · = c × y + c + γy,with vectors c, c and a real number γ.◮ Use 7-tuple (c, c, γ) to encode velocity vector fields whichoccur with smooth equiform motions.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Uniform motions and Euclidean invariant surfaces◮ Uniform motions have a constant velocity vector field.◮ The Euclidean ones (ẏ = c × y + c) are translations,rotations, and helical motions.◮ The corresponding invariant surfaces are cylindrical,rotational, and helical surfaces.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Uniform motions and equiform invariant surfaces◮ Uniform motions have a constant velocity vector field.◮ The truly equiform ones (ẏ = c × y + c + γy) are exponentialscaling and spiral motion.◮ The corresponding invariant surfaces are conical and spiraloidsurfaces.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Line elementsnyn◮ A line element is a line with a point yon it, encoded via a 7-tuple(n, n, ν) ∈ R 7 ,oνn . . . unit direction vector,n = y × n . . . moment vector,ν = 〈y, n〉 . . . scalar.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Line elements and equiform kinematicsnyẏ◮ Look for “path normal elements”orthogonal to velocity vectors of asmooth equiform motion.◮ Fact. A line element (n, n, ν) is apath normal element of the velocityvector field ẏ = c × y + c + γy iff¯n = y × n, ν = 〈y, n〉〈n, c〉 + 〈n, c〉 + νγ = 0.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Line elements and equiform kinematicsnyẏ◮ Look for “path normal elements”orthogonal to velocity vectors of asmooth equiform motion.◮ Fact. A line element (n, n, ν) is apath normal element of the velocityvector field ẏ = c × y + c + γy iff¯n = y × n, ν = 〈y, n〉〈n, c〉 + 〈n, c〉 + νγ = 0.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Line elements and equiform kinematicsnyẏ◮ Look for “path normal elements”orthogonal to velocity vectors of asmooth equiform motion.◮ Fact. A line element (n, n, ν) is apath normal element of the velocityvector field ẏ = c × y + c + γy iff¯n = y × n, ν = 〈y, n〉〈n, c〉 + 〈n, c〉 + νγ = 0.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Recognizing invariant surfacesTheoremThe coordinates (n, n, ν) of the normalline elements of a surface fulfill a linearhomogeneous equation〈n, c〉 + 〈n, c〉 + νγ = 0⇐⇒ the surface is invariant w.r.t. theuniform motion determined by thevelocity vector fieldẏ = c × y + c + γy.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Classification of invariant surfaces – the exact case1. Given surface sample points y i with unit surface normalvectors n i , compute corresponding normal line elements(n i , n i , ν i ) with n i = y i × n i , ν i = 〈y i , n i 〉.2. Find all linear homogeneous equationsfulfilled by normal elements.〈n i , c〉 + 〈n i , c〉 + ν i γ = 0.3. Each such (c, c, γ) means a uniform motion which leaves thesurface invariant. Some surfaces are multiply invariant.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Classification of invariant surfaces – the exact case1. Given surface sample points y i with unit surface normalvectors n i , compute corresponding normal line elements(n i , n i , ν i ) with n i = y i × n i , ν i = 〈y i , n i 〉.2. Find all linear homogeneous equationsfulfilled by normal elements.〈n i , c〉 + 〈n i , c〉 + ν i γ = 0.3. Each such (c, c, γ) means a uniform motion which leaves thesurface invariant. Some surfaces are multiply invariant.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Classification of invariant surfaces – the exact case1. Given surface sample points y i with unit surface normalvectors n i , compute corresponding normal line elements(n i , n i , ν i ) with n i = y i × n i , ν i = 〈y i , n i 〉.2. Find all linear homogeneous equationsfulfilled by normal elements.〈n i , c〉 + 〈n i , c〉 + ν i γ = 0.3. Each such (c, c, γ) means a uniform motion which leaves thesurface invariant. Some surfaces are multiply invariant.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Invariant surfaces◮ Classification of invariant surfaces using ẏ = c × y + c + γy:c = 0 c = 0 c ≠ 0 c ≠ 0 c ≠ 0γ = 0 γ ≠ 0 γ = 0 γ = 0 γ ≠ 0〈c, c〉 = 0 〈c, c〉 ≠ 0cylindrical conical rotational helical spiral surf.◮ Multiply invariant surfaces fall into two or more of theseclasses simultaneously.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Multiply invariant surfaces◮ Surfaces invariant with respect to ≥ 2 uniform motions:cone (2) rot. cyl. (2) log. cyl. (2) sphere (3) plane (4)◮ logarithmic cylinder = cylinder with logarithmic spiral as basecurve (does not occur in applications)rot.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Surface recognition using PCA – the real world case◮ Normal line elements of a surface shaped noisy point set forma point cloud (n i , n i , ν i ) in R 7 .◮ Fit a hyperplane 〈c, n〉 + 〈c, n〉 + νγ = 0 by minimizingF (c, c, γ) = ∑ Ni=1 (〈c, n i〉 + 〈c, n i 〉 + ν i γ) 2 .under the side condition c 2 + c 2 + γ 2 = 1.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Surface recognition using PCA – the real world case◮ Normal line elements of a surface shaped noisy point set forma point cloud (n i , n i , ν i ) in R 7 .◮ Fit a hyperplane 〈c, n〉 + 〈c, n〉 + νγ = 0 by minimizingF (c, c, γ) = ∑ Ni=1 (〈c, n i〉 + 〈c, n i 〉 + ν i γ) 2 .under the side condition c 2 + c 2 + γ 2 = 1.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Surface recognition using PCA – the real world case◮ Quality of fit corresponds to magnitude of eigenvalues ofcovariance matrix∑ Ni=1 (n i, n i , ν i )(n i , n i , ν i ) T ∈ R 7×7 .Solution is given by a corresponding eigenvector (c, c, γ).◮ One small eigenvalue → invariant surface.◮ Two small eigenvalues → twofold invariant surface.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Surface recognition using PCA – the real world case◮ Quality of fit corresponds to magnitude of eigenvalues ofcovariance matrix∑ Ni=1 (n i, n i , ν i )(n i , n i , ν i ) T ∈ R 7×7 .Solution is given by a corresponding eigenvector (c, c, γ).◮ One small eigenvalue → invariant surface.◮ Two small eigenvalues → twofold invariant surface.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Axis and center of a spiral surfaceFrom (c, c, γ) compute axis (a, a) and center z ofequiform motion:a = c,a =1γ 2 + c 2 (c2 c − 〈c, c〉c + γc × c),z =1γ(c 2 + γ 2 ) (γc × c − γ2 c − 〈c, c〉c).M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Does nature produce exact spiral surfaces?◮ Given laser scanner data of a sea shell (Turbo Marmoratus)◮ Axes and centers cluster in piecewise reconstruction:M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Reconstruction: Generator curvesShape of invariant surfaces is determined by a generator curve.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Example: Saxidomus NutalliM. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Example: Helix PomataM. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Segmentation procedureRANSAC is used for detecting1. Planar and spherical parts.2. Twofold invariant parts.3. Parts with simple invariance.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Segmentation & Morphology◮ If normal vectors fit into a rotation → surface part withrotational symmetry is detected.◮ Curve-like surface parts where the surface normals fit thesame rotation are removed, by morphological opening.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

Conclusion◮ Line element geometry and equiform kinematics together withnumerical/statistic techniques (PCA and RANSAC) arebeneficial for recognizing, reconstructing, and segmentingspecial 3D shapes.◮ Applications are in engineering, archeology, and life sciences.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry

References◮ M. Hofer, B. Odehnal, H. Pottmann, T. Steiner, J. Wallner:3D shape understanding and reconstruction based on lineelement geometry. Proc. ICCV’05, Vol. 2:1532–1538, 2005.◮ H. Pottmann, M. Hofer, B. Odehnal, J. Wallner: Linegeometry for 3D shape understanding and reconstruction. InT. Pajdla and J. Matas, editors, Computer Vision - ECCV’04,Part I, LNCS 3021, pages 297-309, Springer, 2004.◮ B. Odehnal, H. Pottmann, J. Wallner: Equiform kinematicsand the geometry of line elements. Submitted to:Contributions to Algebra and Geometry.M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry